For materials that exhibit abrupt
failure, the start of slow crack extension will be followed immediately by the
onset of rapid fracture. The residual
strength capability then requires a strict evaluation of the initial flaw sizes
in the structure. The allowable initial
crack length necessary to maintain the required residual strength will be less
than af ; the design limit
load must also be such that the stress level in the structure is less than si , as shown in Figure
4.4.1. The residual strength
diagram can be evaluated as described earlier through the plot of sf vs. ac
using the relationship K = sbÖpa
for the structural geometry of interest and also employing the failure
criterion based on a critical fracture toughness value, Kcr. The margin
of safety as shown in Figure 4.4.1 allows for
undetected cracks or for subcritical crack growth such that the initial crack
size will not become greater than af.
The following example problem is presented to demonstrate the
application of the steps in constructing the residual strength diagram and also
to analyze the structure for its residual strength capabilities. This example demonstrates the basic concepts
involved in the residual strength capabilities of a single load path structure.
4.4.1 Residual Strength of
Center Cracked Panel
Develop the residual
strength diagram for the cracked finite width panel shown here. The panel is 20 inches wide and
0.375 inches thick with a length of 60 inches. The yield strength (sy) for this material is 78 ksi and the fracture toughness (KIc ) is 40 ksi Öin. The inspection procedure is a viual
inspection capable finding a crack (2a) 2 inches long.
For the center-cracked
geometry configuration shown, the stress-intensity factor K expresses by the relationship (see Section 11.3):
Since we have an explicit
expression for K, using the fracture
toughness failure criterion (plane strain), the residual strength diagram can
be obtained directly. The corresponding
= 40 ksi Öin
and W = 20 inch are given as data and
sf can be obtained for any selected crack
length. The sf vs. ac
curve, which is the required residual strength diagram, can now be plotted.
The residual strength sf of the panel can be estimated from the equation
that is described in the following diagram.
From this figure, for the given operating stress level (20 ksi), the
critical crack size a at which
unstable crack extension would occur can be estimated as 1.2 inches. Thus, to avoid a fracture type failure of
the panel, the structure should not develop a crack of this size. Assume that based on an established visual
inspection schedule, the simple rectangular aluminum panel, uniformly loaded in
tension as shown, could develop a 2.0 inch long, central through-the-thickness
crack (normal to loading) before detection.
This crack length is slightly smaller than the critical crack size (2a) of 2.4 (2 x 1.2 inch) under the
operating conditions so that the margin of safety is small when this inspection
process is employed.
Residual Strength Diagram Determining Critical Crack
Size at 20 ksi Operating Level
To establish the required residual strength level to fit the
inspection schedule, the designer must reduce the crack-tip stress-intensity
factor for the same applied load. One
method is to transfer portions of the load to a stiffening member. Another method is to reduce the operating
load level below the failure level corresponding to the inspection crack size,
although this is not always practiced.